In this paper, we consider the existence of strong solutions of the following p(x)-Laplacian Dirichlet problem via critical point theory:
We give a new growth condition, under which, we use a new method to check the Cerami compactness condition. Hence, we prove the existence of strong solutions of the problem as above without the growth condition of the well-known Ambrosetti–Rabinowitz type and also give some results about multiplicity of the solutions.
Qihu Zhang and Chunshan Zhao. "Existence of Strong Solutions of a P(X) -Laplacian Dirichlet Problem without the Ambrosetti-Rabinowitz Condition" Computers & Mathematics with Applications
Vol. 69 Iss. 1 (2015) p. 1 - 12
Available at: http://works.bepress.com/chunshan_zhao/20/